66,187 research outputs found
Facets for Art Gallery Problems
The Art Gallery Problem (AGP) asks for placing a minimum number of stationary
guards in a polygonal region P, such that all points in P are guarded. The
problem is known to be NP-hard, and its inherent continuous structure (with
both the set of points that need to be guarded and the set of points that can
be used for guarding being uncountably infinite) makes it difficult to apply a
straightforward formulation as an Integer Linear Program. We use an iterative
primal-dual relaxation approach for solving AGP instances to optimality. At
each stage, a pair of LP relaxations for a finite candidate subset of primal
covering and dual packing constraints and variables is considered; these
correspond to possible guard positions and points that are to be guarded.
Particularly useful are cutting planes for eliminating fractional solutions.
We identify two classes of facets, based on Edge Cover and Set Cover (SC)
inequalities. Solving the separation problem for the latter is NP-complete, but
exploiting the underlying geometric structure, we show that large subclasses of
fractional SC solutions cannot occur for the AGP. This allows us to separate
the relevant subset of facets in polynomial time. We also characterize all
facets for finite AGP relaxations with coefficients in {0, 1, 2}.
Finally, we demonstrate the practical usefulness of our approach. Our cutting
plane technique yields a significant improvement in terms of speed and solution
quality due to considerably reduced integrality gaps as compared to the
approach by Kr\"oller et al.Comment: 29 pages, 18 figures, 1 tabl
Selective Use of the Executive Immunity Power: A Denial of Due Process?
Attacks on the government\u27s power to grant immunity to cooperative witnesses have been premised on several grounds, including the due process clause of the fifth amendment. It is upon this clause that the United States District Court of the Southern District of New York based a decision that a defendant was denied due process when the government refused to immunize him after granting immunization to its own witnesses. This article examines traditional arguments against challenging a prosecutor\u27s immunity discretion, the procedural and substantive factors necessary in substantiating a defendant\u27s due process claim, and the effect of immunization on the government\u27s burden of proof in future prosecutions
Unital Quantum Channels - Convex Structure and Revivals of Birkhoff's Theorem
The set of doubly-stochastic quantum channels and its subset of mixtures of
unitaries are investigated. We provide a detailed analysis of their structure
together with computable criteria for the separation of the two sets. When
applied to O(d)-covariant channels this leads to a complete characterization
and reveals a remarkable feature: instances of channels which are not in the
convex hull of unitaries can return to it when either taking finitely many
copies of them or supplementing with a completely depolarizing channel. In
these scenarios this implies that a channel whose noise initially resists any
environment-assisted attempt of correction can become perfectly correctable.Comment: 31 page
Experimental Observation of Quantum Correlations in Modular Variables
We experimentally detect entanglement in modular position and momentum
variables of photon pairs which have passed through -slit apertures. We
first employ an entanglement criteria recently proposed in [Phys. Rev. Lett.
{\bf 106}, 210501 (2011)], using variances of the modular variables. We then
propose an entanglement witness for modular variables based on the Shannon
entropy, and test it experimentally. Finally, we derive criteria for
Einstein-Podolsky-Rosen-Steering correlations using variances and entropy
functions. In both cases, the entropic criteria are more successful at
identifying quantum correlations in our data.Comment: 7 pages, 4 figures, comments welcom
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